Modelling Waves in the Ocean - Lancaster University...Modelling Waves in the Ocean Time Series Wind generated waves treated as stochastic process Sampling displacement over time of

Modelling Waves in the Ocean


Jack McGinnSupervisor: Jake Grainger

September 2020


Time Series

Wind generated waves treated as stochastic processSampling displacement over time of some point in the oceanis denoted by stochastic variable X∆ = [Xt∆]t∈Z formingdiscrete time series


Spectral Density

Continuous Spectral Density

f (ω) =1

2π

∫ +∞

−∞c(τ)exp(−iωλ)dτ

Discrete Spectral Density

f∆(ω) =∆

2π

∞∑τ=−∞

c(τ∆)exp(−iωτ∆)

Aliasing

f∆(ω) =∞∑

k=−∞f (ω +

2πk

∆)


Aliasing


Periodogram

Periodogram estimate for spectral density from wave displacementtime series

I (ω) =∆

2πN

∣∣∣∣∣N−1∑t=0

Xt∆exp(−it∆ω)

∣∣∣∣∣2


JONSWAP

JONSWAP model for spectral density of wind generated waves inthe ocean

SG (ω|θ) = αω−rexp

(−rs

(ω

ωp

)−s)γδ(ω|θ)


JONSWAP


Estimating parameters - Whittle Approximation

Whittle Likelihood function

`W (θ|X∆,N) = −∑ω∈Ω

log(f (ω|θ)) +I (ω)

f (ω|θ)

De-biased Whittle Likelihood function, replace f (ω|θ) infraction with,

E [I (ω)] =1

2πRe

(2∆

N−1∑τ=0

(1− τ

N

)c(τ |θ)exp(−iωτ∆)−∆c(0|θ)

)


Simulated Data


Real Data


Real Data continued


Swell


Robustness of Removing swell


Real Parameters


Further Work

Improve method of testing robustness

Improve method for removing swell

Documents

Modelling Waves in the Ocean - Lancaster University...Modelling Waves in the Ocean Time Series Wind generated waves treated as stochastic process Sampling displacement over time of